Subgroups — #26121533

Order 12032.

1422 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689087	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688887	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#407	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	2	yes	—
6	C₂³×C₄₇⋊C₈	—	2	8	no	—
7	C₂³×C₄₇⋊C₈	—	2	2	no	—
8	C₂³×C₄₇⋊C₈	—	2	2	no	—
9	C₂³×C₄₇⋊C₈	C₄	1	4	yes	—
10	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
11	#15689087	—	2	2	no	—
12	#15689087	C₄	1	4	yes	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
14	#15688887	—	2	8	no	—
15	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#407	—	2	2	no	—
18	C₄₇×#407	C₄	1	4	yes	—
19	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	C₂×C₄	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	16	no	—
25	C₂²×C₄₇⋊C₈	—	2	16	no	—
26	C₂²×C₄₇⋊C₈	—	2	8	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	—	2	4	no	—
30	C₂²×C₄₇⋊C₈	—	2	16	no	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₄₇⋊C₈	D₄	1	8	yes	—
34	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
35	C₂⁵×C₄₇	Q₈	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	—	2	4	no	—
37	C₂³×C₄×C₄₇	—	2	4	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₇	D₄	1	2	yes	—
42	C₂³×C₄×C₄₇	D₄	1	2	yes	—
43	C₂³×C₄×C₄₇	D₄	1	2	yes	—
44	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	4	no	—
48	C₂×M₁₆×C₄₇	—	2	4	no	—
49	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₄₇	—	2	2	no	—
51	C₂²⋊C₈×C₄₇	—	2	8	no	—
52	C₂⁴×C₄₇	—	2	8	no	—
53	C₂⁴×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
55	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
57	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	8	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₄₇	#41	1	4	yes	—
83	M₁₆×C₄₇	—	2	8	no	—
84	M₁₆×C₄₇	—	4	2	no	—
85	M₁₆×C₄₇	—	4	2	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	4	4	no	—
88	#10625329	—	2	8	no	—
89	#10625329	—	4	2	no	—
90	#10625329	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	2	32	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	8	no	—
95	C₂×C₄₇⋊C₈	—	2	8	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	4	4	no	—
98	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	8	yes	—
99	C₂³×C₄₇	—	2	32	no	—
100	C₂³×C₄₇	—	2	8	no	—
101	C₂³×C₄₇	—	2	8	no	—
102	C₂³×C₄₇	—	2	4	no	—
103	C₂³×C₄₇	—	2	4	no	—
104	C₂³×C₄₇	—	2	4	no	—
105	C₂³×C₄₇	—	2	4	no	—
106	C₂³×C₄₇	—	2	4	no	—
107	C₂³×C₄₇	#95	1	4	yes	—
108	C₂³×C₄₇	#108	1	8	yes	—
109	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	—	2	8	no	—
121	C₂×C₄×C₄₇	#95	1	4	yes	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#117	1	2	yes	—
128	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
129	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
130	C₂×C₄×C₄₇	#120	1	2	yes	—
131	C₂×C₄×C₄₇	#120	1	2	yes	—
132	C₂×C₄×C₄₇	#121	1	4	yes	—
133	C₂×C₄×C₄₇	#123	1	2	yes	—
134	C₂×C₄×C₄₇	#123	1	2	yes	—
135	C₈×C₄₇	—	4	4	no	—
136	C₄₇⋊C₈	—	2	8	no	—
137	C₄₇⋊C₈	—	4	4	no	—
138	#10723	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	16	no	—
140	C₂²×C₄₇	—	2	16	no	—
141	C₂²×C₄₇	—	2	16	no	—
142	C₂²×C₄₇	—	2	8	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	#341	1	8	yes	complemented
148	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
149	C₂²×C₄₇	#380	1	2	yes	—
150	C₂²×C₄₇	#385	1	2	yes	—
151	C₂²×C₄₇	#385	1	2	yes	—
152	C₂²×C₄₇	C₂×#108	1	4	yes	—
153	C₄×C₄₇	—	2	4	no	—
154	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
155	C₄×C₄₇	#379	1	1	yes	characteristic
156	C₄×C₄₇	#381	1	1	yes	characteristic
157	C₄×C₄₇	#386	1	2	yes	—
158	C₄×C₄₇	#386	1	2	yes	—
159	C₂⁴×C₈	—	47	1	no	—
160	C₂×#407	—	47	1	no	—
161	C₂×#407	D₄₇	1	1	yes	characteristic
162	C₂×#341	—	47	4	no	—
163	C₂×C₄₇	—	2	8	no	—
164	C₂×C₄₇	—	2	2	no	—
165	C₂×C₄₇	—	2	2	no	—
166	C₂×C₄₇	C₂²⋊#95	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#341	1	2	yes	—
168	C₂×C₄₇	#1810	1	4	yes	—
169	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
170	C₂²×M₁₆	—	47	2	no	—
171	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
172	C₂³×C₈	—	47	2	no	—
173	C₂³×C₈	—	47	4	no	—
174	C₂³×C₈	—	94	8	no	—
175	C₂³×C₈	—	94	2	no	—
176	C₂³×C₈	—	94	2	no	—
177	#407	—	2	2	no	—
178	#407	—	47	4	no	—
179	#407	—	94	2	no	—
180	#407	Dic₄₇	1	4	yes	—
181	C₂×C₂²⋊C₈	—	2	2	no	—
182	C₂×C₂²⋊C₈	—	94	2	no	—
183	#341	—	94	8	no	—
184	C₄₇	#10723	1	1	yes	characteristic, complemented, Sylow
185	C₂⁵	Dic₉₄	1	1	yes	characteristic
186	C₂³×C₄	—	2	4	no	—
187	C₂³×C₄	—	2	4	no	—
188	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
189	C₂³×C₄	C₄×D₄₇	1	2	yes	—
190	C₂³×C₄	C₄×D₄₇	1	2	yes	—
191	C₂³×C₄	D₁₈₈	1	2	yes	—
192	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
193	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
194	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	47	4	no	—
200	C₂×M₁₆	—	94	2	no	—
201	C₂×M₁₆	—	94	4	no	—
202	C₂×M₁₆	—	94	4	no	—
203	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	47	4	no	—
206	C₂²×C₈	—	47	8	no	—
207	C₂²×C₈	—	94	16	no	—
208	C₂²×C₈	—	94	16	no	—
209	C₂²×C₈	—	94	16	no	—
210	C₂²×C₈	—	94	8	no	—
211	C₂²×C₈	—	94	2	no	—
212	C₂²×C₈	—	94	4	no	—
213	C₂²×C₈	—	94	2	no	—
214	C₂²×C₈	—	94	2	no	—
215	C₂²⋊C₈	—	2	8	no	—
216	C₂²⋊C₈	—	94	8	no	—
217	C₂⁴	—	2	8	no	—
218	C₂⁴	—	2	4	no	—
219	C₂⁴	Dic₄₇⋊C₄	1	2	yes	—
220	C₂⁴	Dic₄₇⋊C₄	1	2	yes	—
221	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
222	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
234	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
235	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
236	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
237	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
238	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
239	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
240	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₄₇	1	4	yes	—
244	C₂²×C₄	#10625352	1	4	yes	—
245	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
248	M₁₆	—	2	8	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	M₁₆	—	94	8	no	—
252	M₁₆	—	188	2	no	—
253	M₁₆	—	188	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	4	4	no	—
256	C₂×C₈	—	47	8	no	—
257	C₂×C₈	—	94	32	no	—
258	C₂×C₈	—	94	4	no	—
259	C₂×C₈	—	94	4	no	—
260	C₂×C₈	—	94	8	no	—
261	C₂×C₈	—	94	8	no	—
262	C₂×C₈	—	94	4	no	—
263	C₂×C₈	—	188	4	no	—
264	C₂³	—	2	32	no	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	2	8	no	—
269	C₂³	—	2	8	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	4	no	—
272	C₂³	C₄₇⋊#95	1	4	yes	—
273	C₂³	#13093959	1	8	yes	—
274	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
275	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
276	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
277	C₂³	#13094001	1	4	yes	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	C₄₇⋊#95	1	4	yes	—
287	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
289	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
292	C₂×C₄	#13093980	1	2	yes	—
293	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
294	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
295	C₂×C₄	#13093992	1	2	yes	—
296	C₂×C₄	#13093994	1	2	yes	—
297	C₂×C₄	#13093996	1	4	yes	—
298	C₂×C₄	#13094005	1	2	yes	—
299	C₂×C₄	#13094008	1	2	yes	—
300	C₈	—	4	4	no	—
301	C₈	—	94	8	no	—
302	C₈	—	188	4	no	—
303	C₂²	—	2	16	no	—
304	C₂²	—	2	16	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	16	no	—
310	C₂²	—	2	8	no	—
311	C₂²	#15688887	1	8	yes	complemented
312	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
313	C₂²	#15688981	1	2	yes	—
314	C₂²	#15688995	1	2	yes	—
315	C₂²	#15688998	1	2	yes	—
316	C₂²	C₂×#13093959	1	4	yes	—
317	C₄	—	2	4	no	—
318	C₄	C₂²⋊C₄×Dic₄₇	1	2	yes	—
319	C₄	#15688978	1	1	yes	characteristic
320	C₄	#15688984	1	1	yes	characteristic
321	C₄	#15689000	1	2	yes	—
322	C₄	#15689003	1	2	yes	—
323	C₂	—	2	2	no	—
324	C₂	—	2	2	no	—
325	C₂	—	2	8	no	—
326	C₂	#20241610	1	1	yes	characteristic
327	C₂	C₂×#15688887	1	2	yes	—
328	C₂	#20243424	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#407	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁵×C₄₇	Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#407	D₄₇
hypocenter	C₄₇	#10723
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂×#407

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×#15688887	C₂	semidirect
147. C₂²×C₄₇	#341	semidirect
184. C₄₇	#10723	semidirect
311. C₂²	#15688887	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₄₇⋊C₈	2
C₂×#15689087	2
C₂×#15688887	2
C₂×C₄₇×#407	2
#10723	47